Particle-based algorithm for stochastic optimal control

The solution to a stochastic optimal control problem can be determined by computing the value function from a discretisation of the associated Hamilton-Jacobi-Bellman equation. Alternatively, the problem can be reformulated in terms of a pair of forward-backward SDEs, which makes Monte-Carlo techniques applicable. More recently, the problem has also been viewed from the perspective of forward and reverse time SDEs and their associated Fokker-Planck equations. This approach is closely related to techniques used in score generative models. Forward and reverse time formulations express the value function as the ratio of two probability density functions; one stemming from a forward McKean-Vlasov SDE and another one from a reverse McKean-Vlasov SDE. In this note, we extend this approach to a more general class of stochastic optimal control problems and combine it with ensemble Kalman filter type and diffusion map approximation techniques in order to obtain efficient and robust particle-based algorithms.